The inverse Voronoi problem in graphs
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We introduce the inverse Voronoi diagram problem in graphs: given a graph G with positive edge-lengths and a collection U of subsets of vertices of V(G), decide whether U is a Voronoi diagram in G with respect to the shortest-path metric. We show that the problem is NP-hard, even for planar graphs where all the edges have unit length. We also study the parameterized complexity of the problem and show that the problem is W[1]-hard when parameterized by the number of Voronoi cells or by the pathwidth of the graph. For trees we show that the problem can be solved in near-linear time and provide a lower bound of Ω(n n) time for trees with n vertices.
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